The extension is conservative: every equational theory yields a partial equational theory such that the categories of models coincide, even though our models are in \(\mathsf {Par}\)  rather than in \(\mathsf {Set}\) . The recently proposed Frobenius theories [1]} take their models in the category of relations \(\mathsf {Rel}\) , and are guided by the structure of cartesian bicategories of relations [2]}. Every partial equational theory yields a Frobenius theory and again, the categories of models coincide. We feel that our notion is a sweet-spot. First, we have shown that our notion of partial theories is expressive, capturing a number of important examples. Second, we retain much of the richness of the semantic picture, via a canonical variety theorem and existence of free models.
